Fully Coupled Cardiac Electromechanics with ...

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ScienceDirect Procedia IUTAM 12 (2015) 124 – 133

IUTAM Symposium on Mechanics of Soft Active Materials

Fully Coupled Cardiac Electromechanics with Orthotropic Viscoelastic Effects Barıs¸ Cansıza , Hüsnü Dalb , Michael Kaliskea,∗ a Institute b Department

for Structural Analysis, Technische Universität Dresden, 01062 Dresden, Germany of Mechanical Engineering, Middle East Technical University, TR-06800 Ankara, Turkey

Abstract The objective of this work is to reveal the influence of the experimentally observed passive viscous behaviour of the myocardium on the electromechanical activity by making use of computational approaches. For this purpose, we adopt the fully implicit finite element framework and the passive response is described by the orthotropic viscoelastic material model. The capabilities of the proposed model are assessed by comparing finite element simulations of spiral waves in a heart tissue for the elastic and viscoelastic formulations. The results obtained indicate that rate effects in the passive myocardium play a significant role on the active myocardium response by decreasing the electrical wave speed which consequently effects the evolution of spiral waves. We further investigate the influence of viscosity on the defibrillation phenomenon by means of a three-field coupled finite element formulation of bidomain electromechanics. © by by Elsevier B.V.B.V. This is an open access article under the CC BY-NC-ND license c 2014 2014The TheAuthors. Authors.Published Published Elsevier (http://creativecommons.org/licenses/by-nc-nd/3.0/). Peer-review under responsibility of Konstantin Volokh and Mahmood Jabareen. Peer-review under responsibility of Konstantin Volokh and Mahmood Jabareen.

Keywords: coupled cardiac electromechanics; viscoelasticity; orthotropy; finite element method; bidomain model; monodomain model

1. Introduction The heart is the central unit of the cardiovascular system and its functionality possesses vital importance for human life. Heart disease is considered as a major cause of death in modern life and its treatment requires intense care resulting in high financial cost. Therefore, modeling of complex mechanisms leading to the functioning of the heart has been an active field of research since decades with aim to develop more effective treatment methods. Advent of high performance computing facilities along with difficulties associated with in vivo experiments motivate the development of computational models in order to gain a superior insight into the electromechanical response of the heart. The heart performs as an electromechanical pump which continuously circulates the blood throughout the body and this functionality is achieved by the coupling between electrical wave propagation and the mechanical deformation. Both phenomena by themselves as well as their mutual interaction are of large research interest to scientists. Cardiac cells are excited upon a change of ion concentrations across the cell membrane and two different types of ∗

Corresponding author. Tel.: +49 351 463-34386 ; fax: +49 351 463-37086. E-mail address: [email protected]

2210-9838 © 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).

Peer-review under responsibility of Konstantin Volokh and Mahmood Jabareen. doi:10.1016/j.piutam.2014.12.014

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Barış Cansız et al. / Procedia IUTAM 12 (2015) 124 – 133

approaches exist for the description of this phenomenon; ionic and phenomenological models. Ionic models describe the evolution of individual ion concentrations inside the myocardium that contribute to excitation and are derived from experimental observations on molecular scale 3,21,22,23 . This approach is computationally challenging but necessary, e.g., to simulate drug applications. On the other hand, phenomenological models capture the fundamental characteristics of the myocardium without any molecular description and can be efficacious, e.g., to study the action potential duration and the conduction velocity since it requires less computational effort 24,25 . Furthermore, regarding large deformations and the fibrous nature of the myocardium, anisotropic hyperelastic constitutive formulations have been developed for the passive mechanical behaviour of the myocardium 9,12,13,14,15 . The effect of electrical activity on the mechanical response is, however, described either by an active stress model where the active part is directly generated by transmembrane potential 15,16,17 or by an active strain model that multiplicatively splits the deformation gradient into active and passive parts 8,18,19,20 . Orthotropic time-dependent behaviour of the passive heart tissue has been well documented in the literature 7 . However, the effect of this phenomenon on the coupled cardiac electromechanical response has not been considered in detail up to date. In this contribution, the effect of viscosity on the electromechanical response of the myocardium is investigated. To this end, we utilize the fully implicit finite element framework which strongly couples the mechanical and electrophysiological problem of the myocardium in a mono- and a bidomain setting, respectively 1,6 . Furthermore, the mentioned framework is furnished with the generalized Hill model having the multiplicative decomposition of the deformation gradient into active and mechanical parts. In line with the kinematic split, the free energy function is additively decomposed into active and passive contributions 8. In order to take viscous effects into account, the recently developed orthotropic viscoelastic material model for the passive myocardium, which considers different relaxation mechanisms for the different orientation directions, is consistently embedded in this framework 4. Variations arising due to viscous effects are discussed by comparing the results of simulations with the elastic formulation. For this purpose, we have carried out two types of three-dimensional finite element analyses illustrating defibrillation phenomenon in a rectangular piece of heart tissue with the bidomain formulation and a spiral wave evolution in a generic biventricular heart model with the monodomain formulation. 2. Continuum modeling of cardiac electromechanics In this section, we present the basic kinematics of non-linear continuum mechanics and the differential equations governing the mechanical response and the electrophysiology. For further details about the weak formulation, the consistent linearization and the discretization, see references 1,5,6 . 2.1. Kinematics Let B0 ∈ R 3 and B ∈ R 3 denote the excitable body at initial time t0 ∈ R and current time t ∈ R, respectively. In large strains, the macroscopic motion of the body is described by the non-linear deformation map B0 → B, ϕt (X) := (1) X → x = ϕt (X) that maps at time t ∈ R the material points X ∈ B0 onto spatial points x = ϕt (X) which is also called the current configuration of the material point. The deformation gradient F := ∇ X ϕt (X) maps the unit tangent of the reference or the Lagrangian point onto its counterpart in the current or the Eulerian configuration. Moreover, the Jacobian J := det F > 0 denotes the volume map of a unit reference volume onto its associated spatial counterpart. The quasiincompressible formulation of the cardiac tissue necessitates a decoupled representation of the material response in spherical and volume preserving parts which is achieved by multiplicatively splitting the deformation gradient into volumetric and unimodular contributions Fvol := J 1/3 1

and

¯ := F−1 F . F vol

(2)

In addition, we introduce the symmetric right Cauchy-Green tensor and its unimodular counter part as a strain measure ¯ T gF ¯, C := FT gF and C¯ := F

(3)

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¯ = ∇ X ϕt (X) F

X

B0

¯a F

x

¯m F

B

x

Bi ¯ a and Fig. 1. Illustration of the motion of the continuum body and multiplicative decomposition of the isochoric deformation gradient into active F ¯ m . B0 , Bi and B denote initial, intermediate and current configuration, respectively. mechanical parts F

where the covariant Eulerian metric tensor gab = δab is used for index raising and lowering purposes in the Cartesian framework. The invariant theory of hyperelasticity based on the material objectivity and the principle of material frame invariance requires three isotropic invariants ¯ I1 := trC,

I2 :=

1 ¯ 2 2 (trC) − tr(C¯ ) , 2

I3 = detC¯ = 1.

(4)

Moreover, the contraction of the myocardium is characterized by assuming the existence of an intermediate state between the reference and current configurations whereupon the isochoric deformation gradient (2)2 is multiplicatively ¯ m and the active part F ¯ a 26 , see Fig. 1, decomposed into the mechanical part F ¯ a. ¯ =F ¯ mF F

(5)

The active part of the isochoric deformation gradient is prescribed in terms of the stretch in fiber direction arising due to the myocardial contraction that is merely governed by the transmembrane potential Φ. m For convenience, we further define the mechanical part of the isochoric right Cauchy-Green tensor (3)2 as C¯ := T ¯ m . In what follows, ∇[•] = d[•]/dx, div[•] = ∇[•] : I and [•] ˙ = d[•]/dt symbolize, respectively, the gradient ¯ m gF F and the divergence with respect to spatial coordinates x and the material time derivative of a quantity throughout the manuscript. 2.2. Governing equations Regarding the coupled electromechanical features of the heart tissue, we assume the existence of three independent field variables State(X, t) := {ϕ(X, t), Φ(X, t), Φe (X, t)} , (6) where ϕt , Φ and Φe are, respectively, the deformation map, the action potential and the extracellular potential. The action potential or the so-called transmembrane potential, which is liable for the contraction of myocard tissue, is expressed as the potential difference between the intra- and extracellular media Φ := Φi − Φe , where Φi is the intracellular potential. In the context of the bidomain formulation, cardiac electrophysiology is formulated by means of the coupled diffusion-reaction equations of the intra- and extracellular media which interact through voltage-gated ion channels and ion exchangers.

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2.2.1. Cardiac mechanics A static equilibrium state of the myocardium, which is assumed to be a continuum, can be expressed by conservation of linear momentum J div J −1 τ + b = 0, (7) where τ and b are the Kirchhoff stress tensor and the volume-specific body forces in the reference configuration, respectively. The balance of linear momentum (7) is subjected to the essential and natural boundary conditions ϕ = ϕ¯

on ∂Bϕ

and

t = ¯t

on ∂Bt ,

(8)

with the surface stress traction vector ¯t which can be expressed as ¯t := J −1 τ · n by means of the Cauchy stress theorem where n indicates the outward surface normal on ∂B. Note that the conditions ∂B = ∂Bϕ ∪ ∂Bt and ∂Bϕ ∩ ∂Bt = ∅ have to be satisfied by the surface subdomains. 2.2.2. Cardiac electrophysiology Electric activity of the cardiac tissue is described by the bidomain equations which consider the intra- and extracellular potentials as independent field variables ˙ = J div(J −1 Di · ∇Φ) + J div(J −1 Di · ∇Φe ) + F φ , Φ 0 = J div(J −1 Di · ∇Φ) + J div(J −1 D · ∇Φe )

(9)

with D = Di + De where Di and De are, respectively, the intra- and extracellular normalized conductivities. Furthermore, F φ characterizes the current flow through the cell membrane. The bidomain formulation represents a homogenization of the intra- and extracellular media and the idea behind it is to consider uncoaxial conductivities for the intra- and extracellular domains which enables us to study the electrical activity of the myocardium if an externally applied shock exists. e.g., in case of defibrillation. The bidomain equations are subjected to the essential and natural boundary conditions ¯ Φ =Φ on ∂BΦ and q = q¯ on ∂Bq , (10) ¯ e on ∂BΦe and qe = q¯ e on ∂Bqe , Φe = Φ where the transmembrane and extracellular fluxes are, respectively, expressed as q = D · ∇Φ and qe = De · ∇Φe , along with complementary ∂B = ∂BΦ ∪∂Bq , ∂B = ∂BΦe ∪∂Bqe and disjoint characteristics ∂BΦ ∩∂Bq = ∅, ∂BΦe ∩∂Bqe = ∅ of the subdomains. Remark: Supposing that the intra- and extracellular conductivities are proportional to each other, e.g. Di = αDe where α is a scalar, then (9)1 is reduced to the monodomain formulation while (9)2 becomes redundant ˜ · ∇Φ) + F φ , ˙ = J div(J −1 D Φ

(11)

˜ = Di ( Di + De )−1 De , for further details see reference 5 . in terms of the effective monodomain conductance D 2.3. Constitutive equations Finding out the electromechanical state of the cardiac tissue through Eq. (7) and (9) entails constitutive relations describing the stress response τ, the current source F φ and the conductivities D and Di . The cardiac muscle is contracted in response to propagating depolarization waves. Furthermore, a number of experimental studies designates the existence of adverse effects, namely, deformation of the myocardial tissue can effect the permeability of the ionic channels which ultimately alters the electrical activity of the heart. In view of this coupled nature of the cardiac tissue, the constitutive equations are enhanced by additional terms which enable the mutual interaction between deformation and electric fields.

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2.3.1. Stress response Regarding the quasi-incompressible behaviour of the cardiac tissue, we additively decompose the free energy function into volumetric and isochoric parts ¯ g; F) ¯ Ψ = U(J) + Ψ(

(12)

where the former enforces the incompressibility through the bulk modulus κ U(J) = κ(J − lnJ − 1),

(13)

and the latter depends solely on the unimodular part of the deformation gradient. Accordingly, the additive form of the free energy function leads to the decoupled stress response through the Doyle-Ericksen formula τ = 2∂ g Ψ = p g−1 + τ¯ : P

(14)

¯ g; F) ¯ in the isochoric strain-space and the isochoric in terms of the pressure p := JU (J), Kirchhoff stress τ¯ := 2∂ g Ψ( projection tensor P := I − 13 g−1 ⊗ g−1 , where I := −∂ g g−1 is the symmetric fourth-order identity tensor having the indicial representation Ii jkl := 12 (gik g jl + gil g jk ). Following 8, we further split the isochoric free energy function into the passive and active parts ¯ +Ψ ¯ a ( g; F ¯ m) ¯ =Ψ ¯ p ( g; F) Ψ

(15)

such that the passive response is governed by the total unimodular deformation gradient, while the active response is dependent on the mechanical part of the unimodular deformation gradient. Hence, we write the stress expression associated with the isochoric deformation space in terms of passive and active contributions τ¯ = τ¯ p + τ¯ a .

(16)

Passive stress In oder to model the passive myocardium, we make use of the orthotropic viscoelastic material model that was developed in our recent work 4 . To this end, we first introduce the invariants and the logarithmic strains describing the deformation state of the orthotropic microstructure of the myocardium I4 f := f 0 · C¯ f 0 , ε f :=

1 ln(I4 f ), 2

¯ 0, I4s := s0 · Cs ε s :=

1 ln(I4s ), 2

¯ 0, I4n := n0 · Cn εn :=

1 ln(I4n ) 2

¯ 0, I8 f s := f 0 · Cs (17)

associated with three unit vectors f 0 , s0 and n0 . The passive free energy function is further split into elastic and viscous parts ¯ f 0 , s0 , n0 , I ) = Ψ ¯ e (I1 , I4 f , I4s , I8 f s ) + Ψ ¯ v (ε, I ) ¯ p (C, (18) Ψ where ε := {ε f , ε s , εn } is the set of strains in the logarithmic strain space and I := {α f , α s , αn } indicates the set of strain-like internal variables describing the inelastic deformations due to the rate-dependent material response. For the ground-state elastic response of the constitutive law, a Fung-type free energy function 9 is utilized ai ¯ e (I1 , I4 f , I4s , I8 f s ) = a exp[b(I1 − 3)] + Ψ {exp[bi (I4i − 1)2 ] − 1} 2b 2b i i= f,s (19) afs {exp[b f s I82 f s ] − 1} + 2b f s where a, b, a f , b f , a s , b s , a f s and b f s are non-negative material constants. Furthermore, the non-equilibrium part of the free energy function is assumed to consist of three distinct parts corresponding to fiber, sheet and normal directions and has a simple quadratic form in the logarithmic strain space ¯ v (ε, I ) = 1 Ψ μi (εi − αi )2 (20) 2 i= f,s,n

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governed by the material parameter μi , namely the non-equilibrium shear modulus in fiber, sheet and normal direction, respectively. The decoupled representation of the passive free energy function leads stress to have elastic and viscous contributions as well τ¯ p = τ¯ e + τ¯ v (21) whose algorithmic treatment and explicit expressions can be found in reference 4. Active stress The active free energy, which is stored in the intermediate configuration, is assumed to be governed by the excitation-induced deformation of the myocardium in fiber direction and has the following invariant based form m

¯ )= Ψa ( g; F

1 m η(I − 1)2 2 4f

(22)

m in terms of the material constant η and the mechanical invariant I4mf := f 0 · C¯ f 0 . Bearing in mind the relation in Eq. (5), the mechanical part of the isochoric deformation gradient can be obtained as −1

¯m = F ¯F ¯a F

with

¯ a := I + (λa − 1) f 0 ⊗ f 0 F

(23)

where λa is the stretch generated in the course of contraction and related to the normalized intracellular calcium concentration c¯ through the phenomenological expression 10 λa =

ξ λa 1 + f (¯c)(ξ − 1) min

(24)

in terms of the functions f (¯c) :=

1 1 + arctan(β ln c¯ ) 2 π

and

ξ=

f (¯c0 ) − 1 f (¯c0 ) − λamin

(25)

where β is the material parameter and λamin is the minimum stretch that is attained when the myocardium is in a fully contracted state. The normalized calcium concentration is treated as a local variable and its evolution is related to the non-dimensional transmembrane potential φ c˙¯ = q φ − k¯c

with

c¯(t0 ) = c¯ 0

(26)

in terms of the material parameters q and k. The computation of the current c¯ value is achieved through the implicit Euler scheme in an incremental interval Δt = t − tn c˙¯ ≈

c¯ − c¯ n = qφ − k¯c. Δt

(27)

Linear dependency of Eq. (27) on φ enables to evince a closed form expression of the normalized calcium concentration in the current time step qΔtφ + c¯ n c¯(φ) = . (28) 1 + kΔt Once the updated value of c¯ is computed, one can obtain the mechanical part of the isochoric deformation gradient through Eq. (23) ¯m = F ¯ + 1 −1 F ¯ f 0 ⊗ f 0. F (29) λa Eventually, the knowledge of Eq. (29) enables us to calculate the active contribution of stress in isochoric deformation space (16) (30) τ¯ a = 2η (I4mf − 1) m ¯ m f0 ⊗ F ¯ m f 0. with the second order structural tensor m := F

130


2.3.2. Current source Following the context of phenomenological electrophysiology, we first introduce the non-dimensional transmembrane potential φ and the non-dimensional time τ φ=

Φ + δφ βφ

and τ =

t βt

(31)

in terms of the scaling factors βφ , βt and the potential difference δφ . The normilzed source term f φ is described by Fitzhugh-Nagumo-type excitation equation and assumed to have two contributions f φ = feφ (φ, r) + fmφ ( g; F, φ)

(32)

where ions flowing in and out of the transmembrane govern the purely electrical part feφ (φ, r) and the mechanical part fmφ ( g; F, φ) is generated by the deformation that results in additional ion transmission through ion channels. The former is phenomenologically formulated by means of an ordinary differential equation 25 while the slow recovery variable r chiefly controls the repolarization attributes of the cardiac cell in a local sense feφ = ∂τ φ = cφ(φ − α)(1 − φ) − rφ + I, μ1 r [−r − cφ(φ − b − 1)] ∂τ r = γ + μ2 + φ

(33)

in terms of material parameters c, α, γ, μ1 , μ2 , b and externally applied stimulus I. Hence, the recovery variable r is considered as an internal variable in our algorithmic treatment. Furthermore, the latter term in Eq. (32)2 is associated with the stretch in fiber direction λ¯ fmφ = G s (λ¯ − 1)(φ s − φ) (34) in terms of the maximum conductance G s and the resting potential φ s of the deformation induced ion channels 27 . This contribution to the current source term is activated only if myofibers are under tension λ¯ > 1. Finally, the physical source term can be obtained through the following conversion formula Fφ =

βφ φ f . βt

(35)

2.3.3. Conductivity The propagating nature of electrical waves designates the presence of conductance through a 3D medium in the myocardium tissue which is mathematically indicated as Da = d a f ⊗ f + d⊥a ( g − f ⊗ f )

(36)

for the intra- and extracellular domains {i, e}. For the sake of convenience, we further express the effective conductance for monodomain simulations ˜ = d˜ f ⊗ f + d˜⊥ ( g − f ⊗ f ). (37) D Subscripts and ⊥ indicate along and across the fiber directions, respectively. Note that, Eq. (36) and (37) are deformation dependent second order tensors. 3. Numerical examples A spiral wave is a self sustained rotating wave propagation and leads to unsynchronized contraction of the heart. If it is not immediately terminated, the result is mostly cardiac arrest or damage of the heart tissue. In this section, we carried out two different three-dimensional fully coupled electromechanical simulations in order to present the effect of viscosity on the electromechanical response of the myocardium by means of spiral wave evolution. In the first example, a slice of cardiac tissue having fiber orientation as described in 28 Fig. 6 is discretized into 33 x 33 x 2

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eight-node brick elements. The geometry, initial and boundary conditions of both examples and the discretization and fiber orientation of the second example can be found in 6 . The passive material parameters are adopted from 4 . The rest of the material parameters are presented in Table 1 except for G s = 10 [-], η = 2 kPa and G s = 1.5 [-], η = 5 kPa in the first and second example, respectively. The conversion parameters are chosen as βφ = 100 mV, δφ = −80 mV and βt = 12.9 ms. Furthermore, we employed the Q1P0 element formulation in the simulations. Table 1. Material parameters. Excitation [-]

α = 0.01

b = 0.15

c=8

γ = 0.002

Conduction [mm2 /ms]

d i = 2

i = 0.5 d⊥

d e = 2

e =1 d⊥

μ1 = 0.2 d˜ = 1.6

k = 0.3

c0 → 0

Active response [-]

λamin = 0.65

β=3

q = 0.6

μ2 = 0.3 d˜⊥ = 1.5

φ s = 0.5

In the first example, Fig. 2, as the electrical wave propagates, the viscoelastic sheet deforms relatively less than the elastic sheet as a result of extra constraints caused by viscous effects. Another remarkable peculiarity is the faster wave propagation in the elastic plate (see the snapshots at times t = 120 and t = 480 ms) which ultimately changes the outcome of the simulations. At time t = 460 ms, an external stimulus I = 2.5 is applied for 20 ms to a small rectangular region in order to initiate the spiral wave and we wait for a while to see whether the spiral waves are self sustained. Then, we apply an external shock with q · n = 27.5 mV mm/ms at time t = 2000 ms for 15 ms (see the colour change) to the outer nodes of the plates in order to terminate the unsynchronized and self sustained electrical activity (defibrillation). The idea behind the defibrillation is to eliminate the unsynchronized wave propagation and allow the heart to retrieve its regular contraction. In the viscoelastic plate, the spiral wave vanishes upon the applied shock. However, the magnitude of the shock is not enough to destroy the irregular electrical activity if the formulation is purely elastic. In the second example, Fig. 3, the spiral waves are initiated by applying an external stimulus I = 30 to a rectangular region at time t = 485 ms for 20 ms. Eventually, unsynchronized wave propagation in the viscoelastic heart vanishes. However, a self sustained spiral wave is observed in the elastic heart. The wave speed plays a crucial role on this mechanism. After a while, we observe that the front regions of the spiral wave in the elastic heart are already in relaxed state, while these regions are still in contraction in the viscoelastic heart because of slower wave propagation. Bear in mind that the myocardium tissue can be re-exited if only the transmembrane potential is under a threshold value (relaxed state). Therefore, the spiral wave in the elastic heart can propagate further and becomes self sustained, whereas this is not the case if viscous effects are taken into account.

-20

Φ [mV]

80

t =120 ms

t =480 ms

t =1400 ms

t =2015 ms

t =2875 ms

Fig. 2. Defibrillation of spiral wave by using the bidomain model. First and second rows show the elastic and viscoelastic formulations, respectively.

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t =135 ms

t =505 ms

t =750 ms

t =1160 ms

t =3000 ms

Fig. 3. Comparison of spiral wave dynamics between the elastic and viscoelastic cases by using the monodomain model. First and second rows show elastic and viscoelastic formulations, respectively. The colour scale is the same with in Fig. 2.

4. Conclusion Experimental observations show that the passive myocardium exhibits a rate-dependent behaviour which has an important role on overall functioning of the heart. In order to study this effect, we have combined a recently developed fully coupled implicit finite element framework 6 with an orthotropic viscoelastic material model 4 . The framework enables us to study the viscous effects on the electromechanical response of the myocardium. The electrophysiology of the myocardium is formulated by mono- and bidomain equations and the phenomenological Aliev-Panfilov model is utilized for the description of the current source. On the mechanical side, the myocardium is deemed as quasiincompressible, hyperelastic and orthotropic. Furthermore, orthotropic rate-dependent effects are taken into account by assigning different relaxation mechanisms for fiber, sheet and normal direction. In order to study the effect of viscoelasticity on the electromechanical response of the heart, we analysed the defibrillation phenomenon in a square sheet of a heart tissue and spiral wave evolution in a biventricular heart model. The simulations demonstrate that compared to the purely elastic formulation, the viscous effects cause less deformation and slower wave propagation which results in a self terminated spiral wave and successful defibrillation in the first and second examples, respectively.

Acknowledgement The financial support of the German Research Foundation (DFG) under grant KA 1163/18 is greatfully acknowledged.

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